Since finishing school, I've used very little of the math I learned in school.
I've used a lot of math. Most of it built on high school algebra and geometry. Does that mean that's all I needed to learn?
No.
The key thing I do use is more ephemeral: Mathematical maturity. In my current job, I use math I never learned in graduate school. I'm able to learn it quickly because I learned a lot of math back then. My math classes were a way to develop mathematical maturity.
Which specific math I learned in graduate school was almost incidental. What I picked up was the ability to learn new math.
In programming, you're doing math whenever you convince yourself that a rewritten piece of code means the same thing as the original (by thinking about it, rather than just running it).
I mean, at a certain point, you're just abstracting logical thinking in general and calling it "math". You can say the same about baking, working out your schedule ahead of time so you can pick up your kids and have time to buy groceries, or trying to sort your playlist of non-album singles in a way so that you optimally position each song in a way that minimizes how jarring the switch from one track to the next is
I feel like the whole point of "math" is to abstract these natural patterns so we can like... write them down and investigate them further. We have a limited mental capacity so we abstract it into a syntax/system of meaning so that we can let the paper or the computer do some of the memory work for us.
If you're doing it in your head, sure that's a skill, but is it really math?
Deduction is used in a lot more than just the classes labeled "mathematics". Which makes one question just to what degree the mathematics classes are even useful. Perhaps deductive reasoning could've been taught to do more useful things than factoring polynomials.
Philosophy (at least in the Analytic tradition) involves a lot of deductive reasoning about non-mathematical things (the main difference from maths being that in maths everyone mostly agrees on the foundations, where in philosophy a lot of the reasoning is conditional: IF you believe X, Y, Z, then Q is also true).
I was referring to axioms when I used the term "foundations" above. Inference rules are largely agreed upon across both maths and philosophy (and they're the same ones in both disciplines).
I’ve heard people say things like, “The world is math”, but it never seemed particularly coherent to me. Sometimes I’d assume they meant, “Inference systems with mathematical languages make useful predictions about my empirical experience.”
But now I’m favoring the interpretation that experience is purely formal/syntactic with no semantic component. There is no additional meaning beyond (or behind) appearance.
There is a notion of meaning in that it encodes possibility. A situation with lots of future possibilities has meaning. A dead-end situation has little or no meaning: stagnation, death, ...
If the formal/syntactic appearance is how leads to new possibilities involving more formal/syntactic appearances, then that is its meaning.
Are you suggesting that some people hold the normative position, “One ought prefer situations with more possibilities to those with fewer, ceteris paribus.”
But isn’t that just some hidden metaphysical structure? That is, it’s unavailable as formal appearance?
Could you give an example of philosophers from the analytic tradition that actually build their arguments in an axiomatic way like many practices in mathematics try to? I feel like I've read more attacks on this very approach than actual examples of this approach (from both analytics folks and continental folks)
Classical logic is related to boolean logic, and has its uses.
But a lot of real world "logic" (decision theory, etc) is statistical in nature, often with a big inductive component and often based on "axioms" that only approximate the real world.
In other words, the field of Analytic Philosophy, when not using the proper amount of math and statistics (especially Bayesian statistics) tend to either lead to doubting everything (if they know what things they do not know) or drawing bad conclusions (if treating their axiomatic assumptions as absolute truths)
They have similar problems with science, and maybe Physics in particular, because of the abstraction of modern physics. By interpreting statements made by some physicist too literally, they may (falsely) end up with conclusions that go way beyond the domain of validity of the original statement.
Imho, Philosophy is fine as a side-project for people in academia, but I think philosphers who do not study math, science, psychology, etc at a level comparable to their philosophy work are at risk of ending up in lala-land.
I would estimate that if someone studies only philosophy in college, they will reach a point after about 1 year where their knowledge of philosophy come to a point that requires more understanding of math, science etc than they had when they started. Kind of like a physicist that doesn't take college level math.
I've been programming since I was 14, and getting paid to do it since I was 15 (I know, not impressively young compared to some folks, but still). Approaching 40 now.
It's never felt like math to me. If it did, I probably wouldn't be able to do it. I'm not especially good at math and feel damn near dyslexic when I try to read math-notation-heavy writing. Programming, meanwhile has always come easily and naturally.
... except the languages that try to look like math. Looking at you, Haskell. I get the concepts just fine, but I can't stand the style.
I think that's totally normal. Many (admittedly, sometimes dubious) studies have made the claim that a background in [linguistics | juggling | playing music | etc] is a better predictor of the likeliness that someone will turn out to be a good programmer than a background in math is
As someone with a background in math, I think that makes total sense. I don't feel like there's any magic reason why practicing reasoning under (one of the many paradigms of) mathematics would automatically carry over to the type of reasoning you practice while programming any more than the previously mentioned exercises.
The only times math has helped me is when the programming I'm doing straight up references mathematical concepts I've practiced with (matrices, group theory, etc). Which is a pretty uncommon occurrence in my current position.
If you want your background in something else to help you with programming I think you'll have to put conscious effort into identifying and using the relevant abstractions and metaphors
A professor in college had two terms: microprogramming and macroprogramming.
Microprogramming was the kind of thing where you're trying to, for example, cut down an assembly loop to run in 6 cycles rather than 8, design a numerical algorithm, an operating system scheduler, or similar. This is very mathematical in style of thinking.
Macroprogramming is what happens when you write a database-backed web application, for example. It's much more linguistic. It's about understanding other people's code, writing your code to be understood, and gluing a lot of stuff together.
Both have their place. Macroprogramming is taking over now, but I enjoy microprogramming more.
Umm, no; I'm specifically referring to reasoning that a certain clump of symbols has the same meaning/effect as another one by linking semantics with symbol manipulation rules.
Sure, if you solve it with symbol manipulation rules. That's definitely not the general case when I'm comparing two pieces of code to see if they do the same thing.
I think you've now defined every kind of reasoning to be math.
I don't think that's a good way to define it. But if we do define it that way, then saying "x is math" becomes a very weak and much less interesting statement.
> If you're doing it in your head, sure that's a skill, but is it really math?
Is it not? Adding numbers in my head is certainly math. Algebra in my head is certainly math. Why would it stop being math just because I'm doing it more abstractly?
If you role play in your head the setting of a table, you may be able to count the plates used. This does not imply that you must do the symbolic reasoning of this in an equation.
Yeah I think you are hitting the on what I think is math reduction thinking. That formalized process of thinking is the esencse of what math is. It's quantitative instead of qualitative. Before math formalized those things it's all an eyeballing and impossible to convey without demonstrations. The simple form of that is for baking but if you gave someone an excel sheet and step by step instructions for building radio antennas they can do it because its been formalized. And this certainly transfers to programming because the writing of a program formalizes the recipe
Advanced math typically starts with logic and structures, not numbers. Even when they study numbers, they tend to be focusing on their qualitative properties: compactness, convexity, primeness, etc.
>I mean, at a certain point, you're just abstracting logical thinking in general and calling it "math".
Which isn't that bad a thing to do.
Many programmers could use more formal logical thinking (or, just, logical thinking to begin with). DailyWTF situations, which most of us have seen in our jobs, are often the case of non applying basic logical principles properly.
The point of the article isn't that all symbolic thinking is pointless—just that the specific types of symbolic thinking we spend years teaching aren't actually used very often. Why not spend that time playing with code instead if you want to learn this skill?
I would hesitate to call this math. When baking and you perform substitution of blackberries with raspberries, do you consider that a form of chemistry? It can certainly be justified with it; but I hesitate to say it is the same thing.
Same for when a builder chooses to use a different type of fastener when constructing something. It could be borne of experience, but I would not necessarily call it material engineering at that point. Even if the exact same practice is how materials exploration happens, at a superficial level.
There is also a difference between learning math and understanding math. I was taught the basics of matrix multiplication, finding determinants, solving linear systems of equations and all the stuff in high school. I was half way to my Masters before I really understood what all those things actually where, how they related to different things and all the ways they could be used to actually solve real world problems.
It takes a minimum of about 3 semesters to be able to get linear algebra to the stage of transfer learning. There are slower ways to do it (which most people follow), but there aren't shortcuts -- there is no way high school Algebra 2 can get you there in the time allotted.
That kind of high school surface learning is not a step you can skip, though:
"Surface learning is critical because it provides a foundation on which to build as students are asked to think more deeply."
There are a lot of shortcuts which can cut back on the number of hours, mostly by spreading out the number of years. Early surface exposure helps a lot, as does revisiting topics in different contexts.
I have used very little algebra/calculus/pre-college content, but have used a lot of linear algebra/discrete mathematics. Would this be a common exception to the rule?
I think the author had purely calculus in mind. Discrete mathematics is indeed crucial to understand computer science. Statistics are very important too, especially in our post-fact world, to be able to call out obvious bull**** when you see it.
> Discrete mathematics is indeed crucial to understand computer science.
FAANG engineer here, having worked in multiple companies you would recognize as a principle (or higher) engineer. I've also taken Discrete math when I was a CS undergrad.
I've never used any of the advanced math learned in school, and I've had the pleasure of working on some of the largest and most complex systems ever made. Lots of basic Excel. The "math" I have needed for work (such as TLA+ modeling, percentile distributions, etc) was always learned "on demand".
As a trivial example, to say that you have never used Discrete Math in your work means that you have never for example write if(a && b) or have never done if(!a && !b) and judged it to be cleaner than if(!(a || b)). This also means that you never used a finite state machine, strings, combinatorics, trees, graphs or modulus.
I would believe that you have not explicitly sat down and worked out Discrete Mathematics proofs. But the ideas of Discrete Math are pervasive and unlikely that you have never used them.
I would think that would be more logic than discrete math?
Math is a good abstraction, such that it can be brought to describe the work many other fields do. It doesn't stand that what the others are doing is automatically math, though.
Discrete maths is graph theorems, groups, galois fields, and so on.
There are several algorithms (in for example graphs and crypto) that you can not understand or implement without discrete maths. But even then, only the basic level is required.
Maybe only guy ever needs to know these things in a FAANG? (Meaning, could implementing these algorithms keep more than guy busy?)
GP explicitly stated they were referring to what is taught/learned in a discrete maths class.
finite state machine => usually taught in a theory of computation class
strings => wtf?
combinatorics => would be taught in a discrete maths class but I can totally believe that a software engineer has never used combinatorics aside of solving toy problems
trees, graphs => more likely dealt with in a data structures and algorithms course. That said there is a discrete maths slant on how these topics can be approached, but they involve proofs that nobody really cares about when writing software
modulus => those are usually dealt with under number theory classes, besides, unless you're implementing RSA or some cryptographic function, usually the grade 8 level understanding of modulus (i.e. remainder of a division) suffices
With TLA+ in particular I reckon I could probably teach my parents how to do use it the only maths you really need for simple specifications is being able to think mathematically
That makes sense to me. I find that calculus requires quite a lot of additional knowledge to make it applicable, whereas it's often relatively straightforward to rephrase a problem in terms of linear algebra or discrete mathematics.
He discusses how specificity is key in sports, but actually the consensus now is to avoid specialization for as long as possible. Playing other sports helps you in your primary sport, especially when it comes to injury reduction. The book Range by Epstein covers this well.
The article also seems to be focused on a certain kind of math. Boolean algebra is a type of math that is used regularly in CS and EE. It is so fundamental that if you don’t get it you probably can’t code anything non trivial.
At a bare minimum, excelling in most sports today requires conditioning in the gym. Weight lifting, plyometrics, coordination drills, etc. Cyclists don’t just ride their bikes. Footballers don’t just kick a ball around. Formula 1 drivers don’t just drive their cars.
You can find Nino Schurter’s gym workouts on YouTube (elite mountain biker). Same for some top race car drivers.
This needs more context, otherwise you're just throwing out a very broad opinion. A 12 year old that 'excels' at a sport is definitely not weight lifting, doing plyos, etc... A 15 year old might just be getting into some of those things.
Everything you might be doing in the gym is in support of the task you're trying to accomplish, not the other way around, so I really doubt that the gym is 'the bare minimum' for most athletes.
Sorry, I was thinking about the author’s statement about specificity… if you want to improve your racquetball game, don’t practice squash. And my own experience studying economics and computer science. The math I was required to take (2 semesters calculus, 1 statistics, and 1 discrete math) was in support of the task I was trying to accomplish (basic competence in those subjects). None at a level I would have considered “advanced” beyond the scope of my degree. Friends that studied history or philosophy did even less math than that - many didn’t take any in college (after testing out based on AP exams or similar).
I studied particle physics. The math requirements were similar to what you're listing, though I dabbled slighlty beyond that.
Today, I wish I'd done a lot more math and statistics before focusing on exclusively physics, even if it'd cost me another 1-3 semesters.
I was spending way too much time trying to re-invent the wheel (statistics) in the experimental part, and particle physics really assumed quite a bit more math than what was listed as prerequisites, simply because the university didn't want to delay progression. (To compensate, the bar at the exam was set pretty low.)
I don’t know how much you know about the current state of high level youth sports, but there are many 12 year olds absolutely doing those things. The world of AAU basketball and youth football camps is insane.
AAU basketball literally has leader boards ranking the top 6th and 7th graders in the country. And goes all the way down to 3rd grade.
Okay, my statement was a bit reductive, of course different sports will prescribe different development paths for juniors, but generally speaking, if you're overloading a 12 year old that still has a lot of growing to do, it will end up very poorly(either injury, stunted growth, or burnout).
I did not say anything about juniors being prohibited from competing, healthy competition has immense benefits early on.
When my son was playing youth sports, many of his peers started doing speed/coordination training with a coach in 6th/7th grade. This was usually 1-2x/week and applicable to most sports. Focus was more on first-step speed, running form, and full body movements (jumps, "suicide" sprints/shuttle runs, etc). I thought it was pretty crazy at the time, still do, but at least it was generic.
These days, neighbors have kids in with pitching coaches at age 8, playing travel league baseball 3+ nights/week, plus 2-3 games/weekend. 9 months of the year. That is REALLY crazy to me. How about some swim team, or soccer, or anything else?
Back when I was in high school (mid-90s), the best athletes were all multi-sport athletes. Football/basketball/track-field was a common combo. No idea if that's true today, but I suspect there's a lot more single-sport athletes at the prep level, which is kinda sad.
I know Roger Federer and Nadal were both soccer players until late, and chose to specialize in tennis later down the road. I bet there is some survivorship bias, but a non-negligible number of kids end up hating a sport because of the insane schedules these days and parent oversight. My sister played tennis from the age of 9 until the end of college, and I am not sure she wants to pick up a tennis racquet ever again. I started playing tennis in college, and I am absolutely in love with it, and will be for the rest of my life.
If you mean at the highest levels of sport though, great genetics will completely overwhelm whatever work the less genetically gifted are doing.
This to me is the real problem with over specialization. You have less than gifted parents living out their athletic dreams through their kids but if the parent wasn't that good at the sport there will be correlation with the kid not being that good.
Kids shouldn't specialize in sports too young because there is such a great chance they are not in the right sport at 11.
At the highest highest levels you need genetics and great training. And it also depends on the sport. Super popular sports like basketball and football require both. Niche sports like fencing, you could probably play at a pretty high level with either.
But to play HS sports for example at most schools, either will do the job. And for most people playing the sport they love in high school will be worth it. Even among pro players, their HS experience is often their best experience.
I think even outside of the book, junior athletic development is indeed moving away from specialization at an early age and towards understanding general body movements and coordination abilities early, while having the freedom to play multiple sports(emphasis on play).
I’ve read that the Olympic gold medalist in archery specifically practiced the piano, basketball, and painting just to improve his archery skills. Skills translate.
An article on Russian hockey goalies suggested that the lack of early specialization helped them, and mentioned dance as something they were encouraged to try. (https://www.nytimes.com/2022/06/05/sports/hockey/vasilevskiy... -- sorry about the paywall)
Those activities involve direct muscular and nervous system development around fine motor control of the fingers, which is directly applicable to archery, not some kind of skill translation.
This is a great point. You have to consider directed versus serendipitous need. Most advanced careers are being invented at a rapid pace, and also roles change. We have no idea what math an engineer in 2030 will need to know every day. I have no idea what title I’ll have in 5 years.
Better to know the math and not use it for your particular role, than be unqualified because you can’t
A lot of people wouldn't be able to muster 8th grade math skills and do something usefl with Excel. Maybe there is a non-strawman version of the Math Myth is that the economy would benefit if more people could do that.
American high school education does seem still unfortunately calculus mono-focused, with little of even what makes calculus interesting considered, just ram a bunch of rules for derivation and integration into your head.
A broad base of at least some of stats, linear algebra, and others seems like it would be more useful to most even in STEM to me. I've met folks who were great at those who struggled with geometry and/or calculus, and vice versa. Yet plenty of need for people good at all sorts of those.
Point0. Learning without any memorization… I’m not sure if that is possible. One must hold several concepts inside the head to learn new ones.
Point1. Education system which would strive for bear minimum outcome would be something like “not everybody needs addition, cashiers can take care of that”. (You will cut calculus, somebody else will cut multiplication table)
Point2. which iq range education system is optimised for? (Fair discission; though hard politically). Gifted people are random in population. If you teach and test only the bare minimum content, which 99.9% of population passes with flying colors, then gifted are undertrained and also harder to spot, thus under-utilised.
Point3. Memorization is useful. Even if one understands the concept deeply, the “cheat sheet” inside the head (i.e. memorised info) makes things much much faster. Memorization is abit like muscle-memory and or caching.
I think Point0 and Point3 are currently underappreciated. For all the failings of today’s public education, I believe things used to be “worse” as far as rote memorization went.
The problems with rote learning are obvious, and I don’t mean to dismiss them. But I do think the pendulum has swung so far that today, there is excessive and harmful stigma attached to it. Rote learning is a bit of a leap of faith, and it requires trusting that it will be useful in the end (in other words it comes with an upfront cost)… and this is incredibly powerful for taking some academic shortcuts in a limited amount of time.
I always despised rote learning as a student because I mostly read things once and then remember them. It felt so stupid being forced to solve the same problems again and again. I'd already seen them!
Now, as an adult, I've noticed how much faster I can get past a lot of tricky issues in my work because the patterns I see a lot are familiar and I can glance over them because I know them by heart and don't get stuck on every little detail.
The use of probability distributions and calculus in introductory statistics is a historical artifact. See my comment above about resampling approaches.
> A broad base of at least some of stats, linear algebra
Stats is a non-starter without calculus. You'd just be doing the "ram a bunch of rules for derivation and integration into your head." method since first principles require calculus. Linear algebra is less reliant on calculus, but the fun stuff still requires calculus. I learned basic linear algebra in high school for what it's worth, just enough to solve systems of linear equations.
No, you could focus on the applications of those rules, of which there are more in day to day life. If I see a car's speedometer moving, I don't need to be able to figure out how long it would take to travel a certain distance. But dealing with infrequent events in a smarter way? Super useful.
Teach it the way you suggest and it would just be another useless math class that people would tune out of instead of grounding it in their everyday life experiences.
("Deriving from first principles" isn't the interesting stuff about calculus, either. And we're not requiring quantum physics to derive gravity from first principles in high school...)
Tangentially related to this discussion but I'm curious what you consider "the fun stuff"?
I use linear algebra every day in my work as a graphics programmer. A huge portion of what I do depends on it. While I frequently need calculus it's still a minority of the time.
Derivates obviously show up everywhere: velocity, acceleration, gradients, ...
But it seems to me only the narrow portion relating to physically-based rendering or simulation requires an understanding of integrals. At the moment I can mainly think of how it's used to define continuous functions before introducing the discretization used to actually compute them.
The use of calculus in introductory statistics is a historical artifact. Before computers, it was easier to use mathematical models like probability distributions than to do resampling. With computers, it's easy and more intuitive to use a resampling approach and textbooks that use such approaches are becoming more common. OpenIntro's Introduction to Modern Statistics is a free textbook of that type. You can find it at https://openintro-ims.netlify.app/
I originally studied EE and we did a lot of math. Fourier series, Laplace transforms ... Never used any of it. The most I've used has been some simple statistics and a cubic spline for some curve fitting. Even ML seems to just use basic linear algebra. That said, trigonometry can be quite useful for DIY jobs.
I've used more Geometry & Trig working on my Model Train Set than I have as a professional software engineer. Figuring out table spacing, track radius, overhang from trains, is basic stuff.
I needed to buy a bunch of precisely cut curved wood for a 2nd level (laser cut wood! So Awesome!), and had to actually re-learn the basics of trig in order to provide the details to have everything made.
I too am an EE, though doing digital design. It is true I never have to do a Fourier transform in my professional life, but the time/frequency duality and all the ideas related to it have been useful throughout my career.
In addition, that section of math was the most beautiful thing I've learned in my life.
I find it hard to believe that anyone working as a programmer / software engineer wouldn't be using any actual computer science (as in data structures, algorithm complexity,...). I could believe that most people working as engineers in civil/mechanical engineering would mainly use Excel, but not that they wouldn't be using exponential functions.
But I guess his point is rather that most graduates of engineering schools don't even work as engineers at all, which might be true, I don't know (I even have difficulty believing that given the number of people I know who weren't even trained as engineers and are now working as software engineers).
I don't know, this might not be as far from believing that painters aren't all chemists. Yes, many of them will be mixing paints. No, most of them will not know much more than "these colors mix."
Similarly, is a quarterback running ballistics equations to determine if they can complete a pass? All while running an optimization on the timing of various other players on the field to know if they will have time to make the pass?
It might be true but it is a shallow truth if so. 100% of engineers need to learn elementary algebra even if only 25% of engineers "use" it, for a variety of strong reasons.
Programming is a more interesting issue. If this blog had argued that computer science should be marginalized within software engineering then he might have found something to say (but which has already been said many times by others).
The vast majority of software developers spend their days mapping database tables to web UI's in some way or another.
You don't need any CS or math for that and from experience I can say that most of them don't have this knowledge either.
It's not that those jobs are without challenges but they are in other area's, such as requirements gathering and project management.
In this case, the solid statistics have more to do with the humanities than with the actual mathematics. The math of that kind of stats is fairly easy, but designing a meaningful study of how people learn and apply what they learn requires a ton of work in sociology, psychology, and economics. You have to work around numerous ethical and pragmatic constraints, and deal with millions of confounding variables.
Nice you managed to categorize the work of this person who's been around 20 years longer than Gladwell. The important thing is that you found a way to place yourself above them without ever making a real criticism :)
I disagree with many things he says, but don’t think he can be dismissed so easily. How many other public intellectuals have even attempted, let alone nailed, anything like this?
Bryan Caplan didn't even write the article, Caplan just posted it. The article is by David Edwards, who is a mathematics professor. This is literally the first line of the posted link.
EDIT: I would add that Caplan is a working researcher who has made significant and original contributions to economics, where Gladwell is a journalist who interprets the work of scholars like Caplan. If anything, Gladwell is a Caplan wannabe (although I wouldn't characterize either of them in that fashion, I think they have different goals and methods)
I am one of the exceptions that uses higher level mathematics on a regular basis at work. I agree with his theory that teaching everyone calculus, or preparing to take it, is not necessary for most.
The problem I have is that he is looking at averages. In my experience, there are five tiers: counting, arithmetic, algebra, calculus, and higher math. The USA is simultaneously sending more people to the top two and bottom two, with the bottom two going from a small few to a real population segment. I don't even know if the worst public schools teach anything anymore or if they are just pseudo-prisons to capture taxes.
I won't even go into the woeful state of logical and statistical knowledge. I don't know why those aren't core requirements for graduating high school.
Wow, this is exactly the sort of anti-intellectualist drivel that makes me lose faith in humanity.
And oh, the irony of dismissing the case for mathematics as a thinking aid as “self-serving nonsense” for which the author has “a very low opinion”!
Perhaps if the author had a better attitude towards education and was capable of appreciating the value of mathematics (or more generally, science) in everyday life (to say nothing about the importance of this in democracies), then he would not hold this infantile opinion.
Sorry, not sorry.
Edit: the point is not whether you “use it” when you’re out and about doing your groceries. If you’re judging the value of maths and science by this measure, you’re missing the point, and you’re at such a low level of insight that your perspective is useful to nobody. Rule of thumb: don’t listen to anti-intellectualists.
> Perhaps if the author had a better attitude towards education and was capable of appreciating the value of mathematics (or more generally, science) in everyday life (to say nothing about the importance of this in democracies), then he would not hold this infantile opinion.
Caplan has a doctorate, teaches at a university, has written several books and home educated two of his sons to a level where they published academically before they entered university. He’s also more well read in philosophy and literature than average though that is a low bar to clear.
If you don’t think a professor of Mathematics, David Edwards, understands the value of Mathematics why do you think you do?
My argument is that the value of mathematics is not merely, or even principally, in whether it’s useful in everyday life (the first-order utility), but that it is in fact in how it helps you to think better (the higher-order utility).
The author, as decorated as he may be, seemingly fails to appreciate the latter: in his own words, he has a “very low opinion of this self-serving nonsense”. The exercise of fleshing out the argument against the higher-order value of mathematics, you will note, is left as an exercise to the reader.
This is by definition an anti-intellectualist position: defending such a position, which goes contrary to what should be the default position in general, should require a very high burden of proof, not an offhand dismissal.
No, he really doesn't. His argument against the higher-order value of mathematics is focused on first-order matters, and a vague allusion to an unspecified conflict of interest (?!). This essentially amounts to no more than "brushing off the argument".
> The second argument is the one I always hear around the mathematics department: mathematics helps you to think clearly. I have a very low opinion of this self-serving nonsense. In sports there is the concept of the specificity of skills: if you want to improve your racquetball game, don’t practice squash! I believe the same holds true for intellectual skills. In any case, the case for transference of mathematical skills is unsettled.
You keep on assuming your conclusion and name calling and acting like you’ve made an argument.
I’m sure the author has a high opinion of mathematics as an intellectual pursuit. He is a Math professor. That’s separate from his argument, that it has very limited practical use, even to most engineers and others you would assume would be highly selected for finding it useful.
If Mathematics was enormously useful for teaching argument and precise thought in a reliable way Economics would have eaten all the other social sciences already. Economists know far more Math than the others. Math is uncommonly useful but if it was that good at teaching people how to think, if the transfer of learning argument was true, it would not need to be argued. It b would be bloody obvious.
Please point to me exactly where I did any name calling.
There's this thing called the burden of proof, in philosophy. When you take a difficult position, you must displace this burden. The author has not done so, and it is not my responsibility to show how he is wrong: he has not shown how he is correct.
Allow me to clarify: I do not mean to say that the author is an anti-intellectual person. I have read nothing else that he has written, and I do not know him.
What I am saying (or at least what I intend to say) is that he is defending an anti-intellectualist idea/position, which is just a fact by definition.
> What I am saying (or at least what I intend to say) is that he is defending an anti-intellectualist idea/position, which is just a fact by definition.
Saying something is of limited practical value is not an anti-intellectual position. I do not think most people get any practical value out of the chemistry or physics they learn in school.
The transference of mathematical skill is only "unsettled" in the sense that it is unclear what would it would be transferred to. It is already an entire epistemological category. How much more transference is required?
You’re just assuming your conclusion, that the thing you enjoy and value, intellectual stimulation, should be supported by default. Both of the people you’re dismissing are intellectuals. They’re just self aware enough that they don’t think their preferences should be enforced on others without a rash justification.
Lovers of physical exercise, the production of art or combat can sing the praises of their joys and the benefits according thereto. We don’t rearrange the world to force their interests on the world. If we’re going to do it for our interests we should have actual reasoning behind it, not hand waving.
I understand that the author is "an intellectual", but he is nonetheless defending a fundamentally anti-intellectualist idea.
This is not compatible with an inquisitive, academic or intellectual mindset, no matter that he is a professor or that he holds a PhD. He's free to advance arguments in support of this position, but he should be met with extreme skepticism, as he is defending a difficult position.
> This is not compatible with an inquisitive, academic or intellectual mindset
The world has millions of things to learn about. Why is Math specially subsidized? That’s good argument. That’s not anti-intellectual.
> He's free to advance arguments in support of this position, but he should be met with extreme skepticism, as he is defending a difficult position.
Again, you’re just assuming[edited] your conclusion. You and I and he are people who like ideas and arguing over ideas. He’s arguing that what he teaches isn’t that useful and asking why it gets so many resources. There’s nothing there about the beauty or joy of Math, which I presume he could wax rhapsodic about as a professor. You’re attacking a position that doesn’t appear in article. It’s about the practical value of advanced Math.
> Why is Math specially subsidized? That’s good argument.
First, that's a question, not an argument.
Second, math is special because it's an exact science, as opposed to a natural science. Encompassing the study of "logic", "information", "things that change", "symmetry", "equality" or equivalence... it applies to pretty much everything. In other words it's one of the skills that has the most potential to be widely applicable.
> Again, you’re just aiming your conclusion.
I'm not sure what you mean by "aiming", but I am referring to the burden of proof [1], which the author has not shifted one iota.
Caplan is fantastic at identifying mathurbation in academic navel gazing. His (and the cited author's) opinions are well informed, even if not universal. In economics the concept is called a separating equolibrium
>that it is in fact in how it helps you to think better
And we do use that in everyday life.
Learning higher maths doesn't help us solve problems we encounter all the time or everyday, at least not directly. But it helps us frame them and our thinking conceptually in a quantitative way, and maybe even sometimes in a qualitative one, which is incredibly helpful generally.
Indeed, when someone argues against their own interests we should take them more seriously, not less. That Edwards thinks the main arguments for teaching advanced Math is garbage when that’s his livelihood should make us look at it closely.
> someone argues against their own interests we should take them more seriously, not less.
I'm not sure it is a good advice in general. All the quack "doctors" that go against the mainstream medicine are actually degree-earnt doctors, yet are just scams.
The fact that someone who is a university professor would remove their kids from the general education pool, makes me think they don't really understand one of the main purposes of education.
If their kids were kept in normal schools with other kids, they may have published academically a few years later, but a lot of their peers would have had the privilege of studying with them and improved their academic lives because of that.
> but a lot of their peers would have had the privilege of studying with them
If you need other students to successfully study, that's either yours or the educational system's failure; but other children shouldn't be dragged down for it.
There is a large number of studies on the positive effects of integration in education, from more extreme cases such as not having special schools for disabled children to the more mundane socioeconomic mixing. The consensus seems to be that the positive academic effect a smart kid gets from being put in some advanced school is trumped by the negative effect on the average kids. European countries with a strong egalitarian bend in their schooling system seem to do much better than the American ones and not just on average.
>European countries with a strong egalitarian bend in their schooling system
Europe is not some fantasy-land where all the dreams of the American left come true. Most European countries track students into separate secondary schools based on their demonstrated academic ability.
As an Italian, I know that very well. In my country the high school system very much reflects the socio-economic class of the parents and this separation was designed during fascism and left pretty much unchanged.
When I said "European countries with a strong egalitarian bend in their schooling system" I was thinking more about countries like Finland and Scandinavia rather than implying that all European countries are egalitarian.
> countries with a strong egalitarian bend in their schooling system seem to do much better than the American ones and not just on average.
I'm not sure; France and Russia and their selective systems, respectively classes préparatoires and elite schools, do not seem to do worse than other European countries.
I would agree with you if the author had indicated more clearly that that was his position. He is choosing to defend an extremely difficult position, and his argument is more a rhetorical flourish than anything.
Sorry, just to be clear: it's obvious that this is the position he is taking, but the substance of his reasoning, argument or rationale is lacking. The distinction between the two contrasting positions we're discussing, which are both compatible with the text you highlighted, is what is unclear.
It doesn't matter who he is: the idea he's advancing is fundamentally an anti-intellectual one. It would be an anti-intellectual idea even if it came out of Einstein's mouth.
Edit in response to edit: sorry if my comments give the appearance of anger, that is sincerely not my intended tone. I am admittedly very passionate about the issue of anti-intellectualism in society.
Kids with no aptitude for math are being made miserable for years, some to the point of dropping out of education altogether. Isn’t that anti intellectual? Your blind dogmatism is also anti intellectual.
While it's not the whole story, I've seen a lot more kids struggling with math due to poor teaching (often several years before the class they're currently taking) than due to lack of aptitude.
So, we're trying to optimize a problem that we've built zero infrastructure for? That sounds like paving material for a road no one wants to actually take.
Is it really the teaching which makes the difference though? I struggled with math through all levels of school. As an adult, I struggled with with programming, but I was self taught. After putting in the hours, I eventually learned enough to make a living from it. I then realized, that maybe the students who did well with math put the same effort as I did in learning programming. The difference between math and programming for me is that I immediately saw the utility of learning programming.
In my experience it does. The ways math is taught differs a good bit from when I went to K-12.
In my experience the differences in teachers, motivation, and teaching methods all made a significant difference.
That’s just school, nothing special about Math. The High School Study of Student Engagement has 17% of students bored in every class and 66% bored every day. Most people spend a large part of every day in school suffering. That’s what school is.
You're free to use your own definition for any word you like, but that's not what anti-intellectualism means to the rest of the world.
> Anti-intellectualism is hostility to and mistrust of intellect, intellectuals, and intellectualism, commonly expressed as deprecation of education and philosophy and the dismissal of art, literature, and science as impractical, politically motivated, and even contemptible human pursuits.
Shrug. You're failing to use your intellect to read the whole definition you just copypastad and see how it might relate to what the commenter said: we're often putting primacy on achievement in mathematics as a gate blocking all the other intellectual pursuits mentioned. Not to mention the bias engineering disciplines seem to have, specifically against art and literature as can be witnessed half the time these topics come on this very forum.
I will agree that these things you're naming are "bad things", just like anti-intellectualism is a "bad thing", or that they are counter-productive, or misguided... But they are not anti-intellectual. Can't just lump and confound all the bad categories of things, precision matters.
I think the point they're trying to make is that anti-intellectualism means distrust of all intellectual pursuits and those who pursue them, not some specific ones.
For example, let's say there is a group that believes that only those that study rock music are truly intellectual, and holds them in high esteem, while thinking that all others (including physicists, great writers, prophets etc) are lesser minds and not worth paying attention to. This group is not anti-intellectual, as they hold intellectuals in high regard, though having a wildly off-kilter definition of one.
In contrast, many groups view study itself as a bad activity, and those that practice it as haughty, distant, idealistic, and generally not worth listening to - these groups are what it means to be anti-intellectual; the kind that praise "street smarts" far above "book learning". People advocating for math education as the most important form of human thought are nothing alike (though I do agree they are also wrong, in different ways).
Ahh, yes, the glorification of "street smarts" over the pursuit of knowledge... That really grinds my gears, good example!
I think it's possible for someone to have an anti-intellectual attitude with respect to a subset of domains, however. For example, if I believe there's no real purpose to learning more mathematics, then that's an anti-intellectual attitude towards maths. If I believe there's no real purpose to learning more about music, that's also an anti-intellectual attitude towards the study of music.
(Side note: if I believe there's no use studying homeopathy, well... that's not anti-intellectual, that's just correctly recognizing that homeopathy is not an epistemic domain, ie.: that it is a pseudoscience.)
100% agree. And the fact that you're being downvoted so heavily is pretty indicative of the biases in this community to me. Not that it's a surprise, I have personally struggled against those biases my entire adult life. But it's sad.
We (North America) filter children early in the education system on the basis of their ability to do accurate arithmetic. Most jurisdictions don't even really teach math proper until high school. By the time kids get there, they've lost their self-confidence because the sheer tedium and boredom of the curriculum in the early grades.
Personally: Because of my primary and high school math grades I was barred from entering a computer science degree. But I have a 20+ year software engineering career only because I just really loved computers and pushed my way into the industry, and ended up at Google among other places, despite the lack of degree. But it was hard, really hard, to get into these jobs. And there's always been people in those jobs who gave me the sense they didn't think I belonged there with them.
The way we're teaching math to kids is on the whole really awful. I've seen both my kids struggle with it, despite being gifted intellectually and scoring incredibly high on the WISC-V.
The whole process seems geared towards grinding kids down in repetition and struggle.
Meanwhile we have machines to automate arithmetic and most math generally. In general, my sense of the way math is taught these days: it's an insult to the human spirit.
And people who excel in it and struggled through it and "won" have an intrinsic bias in seeing it as the "right" filter and "right" gating for what they're doing today, even when it clearly is not always directly related.
This is a fantastic comment. I think your analysis at the end is spot-on. It's like the K-12 math curriculum is a hazing ritual for a vast fraternity of "smart people". Without this ritual, their group identity is called into question.
I don't know a single person in this "cabal of smart people" who has a single nice thing to say about the K-12 mathematics curriculum. They largely consider it a travesty, and have vaguely apocalyptic things to say.
> I don't know a single person in this "cabal of smart people" who has a single nice thing to say about the K-12 mathematics curriculum.
You must not know many people. The problem with trying to categorize K-12 mathematics (or other common, but non-uniform experiences) as good or bad or somewhere in between is precisely that the experience is non-uniform. Even being in the same classes as my twin sister (though sometimes not), we formed some wildly different opinions on different teachers and classes and even classmates. The experience was too different even sitting 5' from each other, let alone when speaking about this with people who attended schools in different parts of the country, or even different parts of the same state or city.
I think math suffers from a motivation problem, especially past early arithmetic education.
Reading? Now I have access to all these stories! And it's plainly useful in every single other class, pretty much constantly (even math class—go figure, this math book has more words than numbers in it!). I can read subtitles (on anime, say—it's gone pretty mainstream now, for kids). I can read video games without voice acting, and follow them (Breath of the Wild requires lots of reading, for instance). It is constantly, obviously useful, and enables access to lots of things that most kids want to do.
History? Social studies? Largely an application of reading—and hey, look, more stories! Great! Oh and now I understand that movie I watched better, or that game I'm playing.
Science? Well look at that, a ton more reading, and when it's taught halfway right it's like working through a mystery story, even. Fun!
Math? It really seems like you're just doing it for its own sake, and... that's it. The actual facts or what's intended don't matter—that is what it feels like. It's occasionally useful in other classes, but not constantly like reading is, and often that math is pretty simple compared to the stuff you're learning in math class. Kids aren't dumb and recognize quickly that word problems are contrived, pandering, and ridiculous, and meanwhile have a hard time finding applications for any of this even if they go looking for it. Their parents? They probably see them rely on reading (even just signs or directions) 1000x more often than they see them apply math beyond what's taught by 6th grade. Reading? Useful daily. Math? Only occasionally, and it's mostly the simple stuff.
To a student, it feels like math is about as useful as learning some constructed language that everyone's telling them is super important but in their actual experience sure doesn't seem to be—I mean, no adults they know even read, write, or speak it, and they seem to be getting by just fine. It feels like you're being gaslit or having some kind of practical joke played on you. Of course it's hard to care about it, unless you're the sort who just loves mathematical puzzles, like the way some people are really into crossword puzzles, but most people aren't.
I don't have any kind of solution to this, but IMO it's the fundamental problem with math education.
Reading? Yes, the intricacies of clause structure? No, no one uses that besides a few people.
Science? Almost no one I know uses chemistry or biology. I don’t even remember how to balance a chemical equation.
The same arguments against mathematics education can be used for almost anything.
If “Reading” is the example for English education, then “addition, subtraction, multiplication, and division” are the analogs in math, and they’re used by pretty much everyone in day to day life and in other classes. Hell, they’re even used when reading.
Yes, lots of the memorization-heavy math from 6th grade and earlier is often useful to most people. Nowhere near as constantly in use as reading, but at least a little is used by most people almost every day. Basic arithmetic, trivial calculations involving percentages or fractions, and (far less) simple variable substitution do see actual use by people not in fields that are extremely math-heavy. Maybe the very occasional area/volume formula or pythagorean theorem, if they're DIYers.
The next 6 years? Not so much. Meanwhile, at least the english classes have stories in them, and the kids might not hate all of them.
> Reading? Yes, the intricacies of clause structure? No, no one uses that besides a few people.
Formal grammar is barely taught at all, now. For better or worse. Drilling spelling words has also fallen out of favor.
> The same arguments against mathematics education can be used for almost anything.
They really can't, nowhere near the same degree. If we ignore that and assume more than a tiny minority of kids will give a shit about math if we just present it to them harder or more mathily, then it's not gonna get better.
> Do you ever have a set of facts and reason about what you can conclude from them? That's math.
> Do you ever play board games and try to come up with the best strategy? That's math.
If zero people who aren't math majors think that's true—which is the case—then our 13 years of compulsory math education is to blame for that.
If people manage to do both well enough to get by just fine in life, without any formal math education on it, then one wonders how necessary math education is to effectively do the math that people actually do.
(personally, I'd look in the philosophy section for the former, at least, seeing as all my logic books, except the one specifically about mathematical proofs, are philosophy textbooks, including the ones full of symbols)
Why does this argument not apply to any subject that a number of students find challenging? If the education system were based on this kind of thinking we would basically have no classes.
I certainly have NEVER used biology at work, and I didn’t do well in it in high school or college.
Biology is never touted as something that will be helpful in the workplace. But if you learn what viruses and bacteria are and internalize the fact that we are all made of cells and are immersed in a world of micro organisms, maybe when someone explains to you what a vaccine does or how your diet affects your gut bacteria you will not fall from the sky and claim it's a satanic ploy
Washing your hands in the bathroom is an example of using biology at work. Only recently doctors would refuse to do it before surgery because nobody could explain what it did, so it wasn’t logically correct to do it.
Because for those other subjects it's often correct, so it would be to knock in an open door.
Math has a much stronger claim to be generically useful. Related fields, such as physics may in fact be seen something worth studing more because it teaches you how to apply math, than for the likelihood that you may end up in a job that requires the physics itself.
My own experience is that math/stats is by far the most useful (non language) topic I studied in school/university, even more so that Computer Science (and I work in IT).
Not everything is the US educational system, though. I'm not saying it is necessarily better in some parts of Europe, but I do believe that math education is a basic necessity, and while it is often done badly [1], we shouldn't make it optional.
[1]: In my not-professional experience I see that people get lost in math down the road and will struggle in later classes because they have elementary troubles understanding even the basics of what an equation is. For example, I was baffled when one of my classmates argued that the x and y in an equation can't denote the same value.. But I also experienced the same thing when I tried to teach/help math to my sister. Math is simply not a subject like every other where not remembering the previous year's material is not a problem -- it necessarily builds on itself.
You presume lack of aptitude. I presume the ineptitude of teachers in the vast majority of cases, through about the algebra level, after which case the levels of abstraction do start learning on genuine aptitude for abstract thinking.
That is an education policy issue, not a reason to not teach math as a key skill.
People who don't ever get math can still learn and work to their strengths.
The point of the education system at this point of time ought to be to install a love of learning (and different people will love different subjects) but we are still at the stage of producing workers.
Other kid's lives are being made miserable for years by other subjects. So what is the solution here? Only teach kids things they want to learn?
I mean I personally think school is overrated, especially with high quality YouTube videos available for everything. But I don't think that is a common notion?
Edit: OK, you know what, why not - yes, abolish schools! They are just prisons for kids.
Obviously, the solution is to calibrate how much of each subject we teach to the general student population, and track people with aptitude into deeper iterations of subjects that are less useful for non-specialists. Aspiring lawyers don't need stunt-integration calculus, and aspiring physicists don't need 20th Century American Lit. We have no trouble with this concept in other fields: you don't take AP Studio Art unless you're tracking to a BFA. But we shoehorn everyone into a math sequence that is not especially valuable.
But isn't that what is already supposed to happen? Presumably school authorities are trying to work out a useful curriculum, that they adjust from time to time?
Maybe a more useful discussion would be how to measure success and how to determine what is necessary? This "8th grade maths is enough" is a bit too handwavy.
I can actually imagine that there would not have to be as much focus on doing calculations on paper or in one's head, as everybody has a calculator ready at all times these days. As one example.
But not very specific so far. We only have some anecdotes from that article writer about 8th grade maths.
To start, I think everybody should be able to do financial calculations (like compounding interest), and be able to decipher the charts they are being bombarded with every day in the media.
Other than that I guess I agree with some other posters that it is the people who figure thinks out who are often also good in maths. I am not even sure how to teach that and if traditional maths at school helps.
You asked "what's the solution here". I'm just saying: there are solutions here, and they aren't abstract. Is there a widespread belief that the current US high school math sequence is good? I feel like I've read a lot of different takes, many from mathematicians and scientists, about why it's not good.
The question was in reply to the argument that maths should be abolished or toned down because it makes kids miserable. So if that is the criterion, what about all the other subjects that make kids miserable. I personally enjoyed maths, but other subjects made me miserable.
I don't think that means everything is fine as it is, but also that simply getting rid of subjects because people don't enjoy them may be too much.
Nothing there said it's not valuable. Just that it's not the acme of value for so many things. Critical reading and thinking skills are also at least as valuable.
Not discussed is that statistics and logic are a much better use of instructional time than calculus for most students.
This is a terrible article. Besides the spelling errors and the repetition of "Excel and eighth grade level mathematics" 4 times in the first 4 paragraphs, it seems like a really really pure case of the author doing his best to rationalise an opinion that he's come to through intuition
Can you please not post flamewar comments to HN? You broke several site guidelines in this thread, including the ones that ask you not to call names in arguments, not to snark, and not to fulminate.
But I don't believe there was any name-calling, as I attempted to clarify in another comment: the "anti-intellectualist" label was intended to refer to the idea expressed in the article, not to the person who wrote it. (And even in the case where the label were attached to a person, it's not really an insult: I mean, it's definitely not a compliment, but to be clear, it says absolutely nothing of the person's intelligence, and rather speaks to their attitude towards "the pursuit of knowledge" in the abstract.)
I recognize that my intended meaning was not as clear as it should have been in the root comment, but I hope I have clarified this anywhere else. This was careless phrasing, but there was no ill intent.
I'm sure there was no ill intent! but "this is anti-intellectualist drivel" is name calling in the sense that the HN guidelines use the word; it doesn't have to be personal to qualify:
"When disagreeing, please reply to the argument instead of calling names. 'That is idiotic; 1 + 1 is 2, not 3' can be shortened to '1 + 1 is 2, not 3." - https://news.ycombinator.com/newsguidelines.html
There is a lot of statecraft behind governments promoting this myth, so I wouldn't be reading into it too much. The countries listed want as high as possible of a population of scientists, and they will forecast doom and gloom to try and convince anyone with a passing amount of patriotism to work in those fields. There never was a clear proof that we needed more.
I started a math tools company (mathpix.com) and I could not agree more! US needs humanities far more than STEM. US is extremely strong at STEM and extremely weak at humanities. This includes understanding of the real world and human affairs. Young people now have zero understanding of religion, the military, geopolitics, or really how to think critically about ANYTHING related to social sciences. The truth about the reason for the death of social sciences in the US is a dark and complex one, I’m not even so sure that truth is even important. What matters is what’s next, and it had to start with a little bit of realist humanities education for young people.
A curriculum focused on STEM or a curriculum focused on humanities would do little to change how the younger generation thinks about the world. The reality is that current education patterns are stuck in rigid patterns that confine creativity into a small box for a majority of students. There are rules upon rules upon rules which disable student from thinking critically and creatively.
John Adams, second president of the United States, in a letter written to his wife in May 1780:
> I must study Politicks and War that my sons may have liberty to study Mathematicks and Philosophy. My sons ought to study Mathematicks and Philosophy, Geography, natural History, Naval Architecture, navigation, Commerce and Agriculture, in order to give their Children a right to study Painting, Poetry, Musick, Architecture, Statuary, Tapestry and Porcelaine.
Somewhere along the line it seems we got stuck at what Adams wanted his sons to study, and never really moved on to what he hoped his grandchildren could study.
I've used calculus very little, algebra quite a lot. One of the side-benefits of learning calculus (apart from reading others' work) is how much solving those problems reinforces your algebra skills.
Solving one scary calculus problem thrown at me as an undergrad - an integral - resulted in a solution which broke it into 16 separate integrals. Days of algebra were involved.
You might not be writing out integrals and derivatives by hand, but I think most engineers use calculus a lot more frequently than they realize it.
* If daily user growth is increasing linearly, total user count is exponential over time.
* If I have a radially symmetric shape it's center of gravity is going to be centered on the axis of symmetry - if the density is uniform. But if the density isn't uniform, where will the center of balance shift?
This is single dimension and multi-dimensional calculus, respectively. I bet most engineers use at least the former, at an intuitive level, on a regular basis.
My experience learning math as a programmer was that there was a strange relationship between how much math I understood and how useful I felt math was.
When I didn't understand calculus I thought that was pretty unessential for being a good programmer. Strange thing though, after learning calculus I find myself using to help understand or model problems a couple of times a week at least.
When I didn't understand linear algebra I didn't see how useful it was for anything outside of the obvious niches like graphics programming. But once I learned it I was able to reduce a remarkable number of complex problems to literally 1-3 lines of code.
Even when I knew statistics, I didn't see much use in all that fancy advanced theoretical stuff. Now I find one of the most useful, practical tools when modeling problems is the Laplace approximation, and knowing the Cramer-Rao bound let's me realize the limits of the results I get from this.
Perhaps most famously, the great mathematician G.H. Hardy, in 1940, discussed how permanently impractical and "useless" number theory was (to be clear, he cited this as a virtue of number theory) in his Mathematicians Apology. Just a few years later Alan Turing (and several others) realized that number theory was the key to securing (and undermining) cryptographic systems.
The lesson for me has been that programmers (and I'm sure many other professions) have a habit of finding good uses for the math they've learned.
As a SWE at a FANG, I need to use college level math maybe...2-3 times a year? Almost never. But, when I do need to use it, it is for situations where I would probably never come up with an adequate solution if I hadn't studied those areas of math.
Well...not that I explicitly recognize, but I'm sure it is possible that having been trained in it in the past, I may see certain problems a certain way. I guess I should phrase it like: I learned some math 15+ years ago but then after college I did not continue studying math or even continue refreshing the things I already learned, and I only find myself studying math-related concepts a couple times a year in my job.
Eh, I don't think this has been tested. We've only taught a population to memorize random math functions. We haven't succeeded in widespread understanding of math ever.
The welfare cliff and similar issues wouldn't happen if people could communicate using continuous functions.
> I ... had two students in the class who had been engineers and one who had been an actuary. They claimed never to have used anything beyond Excel and eighth grade level mathematics; never a trig function or even a log or exponential function!
That seems incredible to me. I guess that means I live in a bubble.
Did I miss the thesis? He concludes with a statement that isn’t even true.
Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.
I studied economics and computer science at a top-20 university. I did a grand total of one course in the math department (1st semester calculus). Everything else was specific to my areas of study. I don’t recall anybody being forced to take high level mathematics outside of engineering programs. Even within the hard sciences, the requirement was usually something like “take 2 semesters of math, plus statistics.” Or something like that.
Would the author also argue that since I don’t use history at my job I should never have taken government, architectural history, or a survey of western religions? Those all contribute to my understanding of the world around me. As does math.
If his point is we need more generalists with a broad understanding of the world, including social sciences and history, that could be a valid point, but it’s not the one he made.
My top 20 school had us take 4 semesters of calculus and 2 other math department classes for a cs degree. The econ major only required one semester of calculus. The upper level micro classes ended up being funny because they spent a considerable amount of time teaching multivariable calculus since taking derivatives is 90% of microeconomics. They even spent multiple classes proving lagrange multipliers without ever referring to them by name.
That micro comment sounds about right - my degree didn’t require many specific courses in the math department, but we did plenty of math as part of the program, it was just tailored to the specific thing we were trying to do.
Economists describe this kind of filter (requiring higher math) in terms of separating equilibria for skills. The conceptual idea is that the separation is too costly for the low skilled group, and even a portion of the high skilled group cannot afford to pay the costly achievement.
This is very much in line with Caplan's prior views. It holds true to a degree, but probably isn't universal.
In my engineering degree, the instructor would do the hard math on the board (or hand wave it away), and then often we'd hit the end result with a taylor series expansion or something like that. This gives a nice locally linear model that we could play around with using good old algebra.
Sure -- most engineers shouldn't use advanced math, because that's easy to screw up. But either they are aware that their linear models came from some more complex equations that have been simplified, or they are living in a magic demon-haunted world where equations are just runes handed down by mystic sages. Hopefully it is the former!
Teaching engineers the hard math serves dual purposes -- it helps them understand where their models come from, and somebody is going to have to invent the models. They are one population that has already shown at least some mathematical aptitude.
324 comments
[ 2.9 ms ] story [ 275 ms ] threadI've used a lot of math. Most of it built on high school algebra and geometry. Does that mean that's all I needed to learn?
No.
The key thing I do use is more ephemeral: Mathematical maturity. In my current job, I use math I never learned in graduate school. I'm able to learn it quickly because I learned a lot of math back then. My math classes were a way to develop mathematical maturity.
Which specific math I learned in graduate school was almost incidental. What I picked up was the ability to learn new math.
I feel like the whole point of "math" is to abstract these natural patterns so we can like... write them down and investigate them further. We have a limited mental capacity so we abstract it into a syntax/system of meaning so that we can let the paper or the computer do some of the memory work for us.
If you're doing it in your head, sure that's a skill, but is it really math?
Inference rules, axioms, language?
I’ve heard people say things like, “The world is math”, but it never seemed particularly coherent to me. Sometimes I’d assume they meant, “Inference systems with mathematical languages make useful predictions about my empirical experience.”
But now I’m favoring the interpretation that experience is purely formal/syntactic with no semantic component. There is no additional meaning beyond (or behind) appearance.
If the formal/syntactic appearance is how leads to new possibilities involving more formal/syntactic appearances, then that is its meaning.
But isn’t that just some hidden metaphysical structure? That is, it’s unavailable as formal appearance?
But a lot of real world "logic" (decision theory, etc) is statistical in nature, often with a big inductive component and often based on "axioms" that only approximate the real world.
In other words, the field of Analytic Philosophy, when not using the proper amount of math and statistics (especially Bayesian statistics) tend to either lead to doubting everything (if they know what things they do not know) or drawing bad conclusions (if treating their axiomatic assumptions as absolute truths)
They have similar problems with science, and maybe Physics in particular, because of the abstraction of modern physics. By interpreting statements made by some physicist too literally, they may (falsely) end up with conclusions that go way beyond the domain of validity of the original statement.
Imho, Philosophy is fine as a side-project for people in academia, but I think philosphers who do not study math, science, psychology, etc at a level comparable to their philosophy work are at risk of ending up in lala-land.
I would estimate that if someone studies only philosophy in college, they will reach a point after about 1 year where their knowledge of philosophy come to a point that requires more understanding of math, science etc than they had when they started. Kind of like a physicist that doesn't take college level math.
It's never felt like math to me. If it did, I probably wouldn't be able to do it. I'm not especially good at math and feel damn near dyslexic when I try to read math-notation-heavy writing. Programming, meanwhile has always come easily and naturally.
... except the languages that try to look like math. Looking at you, Haskell. I get the concepts just fine, but I can't stand the style.
As someone with a background in math, I think that makes total sense. I don't feel like there's any magic reason why practicing reasoning under (one of the many paradigms of) mathematics would automatically carry over to the type of reasoning you practice while programming any more than the previously mentioned exercises.
The only times math has helped me is when the programming I'm doing straight up references mathematical concepts I've practiced with (matrices, group theory, etc). Which is a pretty uncommon occurrence in my current position.
If you want your background in something else to help you with programming I think you'll have to put conscious effort into identifying and using the relevant abstractions and metaphors
Microprogramming was the kind of thing where you're trying to, for example, cut down an assembly loop to run in 6 cycles rather than 8, design a numerical algorithm, an operating system scheduler, or similar. This is very mathematical in style of thinking.
Macroprogramming is what happens when you write a database-backed web application, for example. It's much more linguistic. It's about understanding other people's code, writing your code to be understood, and gluing a lot of stuff together.
Both have their place. Macroprogramming is taking over now, but I enjoy microprogramming more.
I don't think that's a good way to define it. But if we do define it that way, then saying "x is math" becomes a very weak and much less interesting statement.
Is it not? Adding numbers in my head is certainly math. Algebra in my head is certainly math. Why would it stop being math just because I'm doing it more abstractly?
Whether it is using symbols, pebbles or categories, it is neither better nor worse.
But they make people think that “maths” is just a terrible abstract construction.
Advanced math typically starts with logic and structures, not numbers. Even when they study numbers, they tend to be focusing on their qualitative properties: compactness, convexity, primeness, etc.
Which isn't that bad a thing to do.
Many programmers could use more formal logical thinking (or, just, logical thinking to begin with). DailyWTF situations, which most of us have seen in our jobs, are often the case of non applying basic logical principles properly.
What's interesting is when the real, practical world runs into the theoretical, abstract math world.
https://petapixel.com/2019/07/05/goodbye-aberration-physicis...
Same for when a builder chooses to use a different type of fastener when constructing something. It could be borne of experience, but I would not necessarily call it material engineering at that point. Even if the exact same practice is how materials exploration happens, at a superficial level.
https://learningcurrents.weebly.com/visible-learning-surface...
It takes a minimum of about 3 semesters to be able to get linear algebra to the stage of transfer learning. There are slower ways to do it (which most people follow), but there aren't shortcuts -- there is no way high school Algebra 2 can get you there in the time allotted.
That kind of high school surface learning is not a step you can skip, though:
"Surface learning is critical because it provides a foundation on which to build as students are asked to think more deeply."
There are a lot of shortcuts which can cut back on the number of hours, mostly by spreading out the number of years. Early surface exposure helps a lot, as does revisiting topics in different contexts.
https://en.wikipedia.org/wiki/Spacing_effect https://en.wikipedia.org/wiki/Spiral_approach
FAANG engineer here, having worked in multiple companies you would recognize as a principle (or higher) engineer. I've also taken Discrete math when I was a CS undergrad.
I've never used any of the advanced math learned in school, and I've had the pleasure of working on some of the largest and most complex systems ever made. Lots of basic Excel. The "math" I have needed for work (such as TLA+ modeling, percentile distributions, etc) was always learned "on demand".
As a trivial example, to say that you have never used Discrete Math in your work means that you have never for example write if(a && b) or have never done if(!a && !b) and judged it to be cleaner than if(!(a || b)). This also means that you never used a finite state machine, strings, combinatorics, trees, graphs or modulus.
I would believe that you have not explicitly sat down and worked out Discrete Mathematics proofs. But the ideas of Discrete Math are pervasive and unlikely that you have never used them.
Math is a good abstraction, such that it can be brought to describe the work many other fields do. It doesn't stand that what the others are doing is automatically math, though.
There are several algorithms (in for example graphs and crypto) that you can not understand or implement without discrete maths. But even then, only the basic level is required.
Maybe only guy ever needs to know these things in a FAANG? (Meaning, could implementing these algorithms keep more than guy busy?)
finite state machine => usually taught in a theory of computation class
strings => wtf?
combinatorics => would be taught in a discrete maths class but I can totally believe that a software engineer has never used combinatorics aside of solving toy problems
trees, graphs => more likely dealt with in a data structures and algorithms course. That said there is a discrete maths slant on how these topics can be approached, but they involve proofs that nobody really cares about when writing software
modulus => those are usually dealt with under number theory classes, besides, unless you're implementing RSA or some cryptographic function, usually the grade 8 level understanding of modulus (i.e. remainder of a division) suffices
The article also seems to be focused on a certain kind of math. Boolean algebra is a type of math that is used regularly in CS and EE. It is so fundamental that if you don’t get it you probably can’t code anything non trivial.
You can find Nino Schurter’s gym workouts on YouTube (elite mountain biker). Same for some top race car drivers.
Everything you might be doing in the gym is in support of the task you're trying to accomplish, not the other way around, so I really doubt that the gym is 'the bare minimum' for most athletes.
Today, I wish I'd done a lot more math and statistics before focusing on exclusively physics, even if it'd cost me another 1-3 semesters.
I was spending way too much time trying to re-invent the wheel (statistics) in the experimental part, and particle physics really assumed quite a bit more math than what was listed as prerequisites, simply because the university didn't want to delay progression. (To compensate, the bar at the exam was set pretty low.)
AAU basketball literally has leader boards ranking the top 6th and 7th graders in the country. And goes all the way down to 3rd grade.
I did not say anything about juniors being prohibited from competing, healthy competition has immense benefits early on.
Key word being playing. It helped me learn teamwork and good sportsmanship, which is supposed to be the point.
When my son was playing youth sports, many of his peers started doing speed/coordination training with a coach in 6th/7th grade. This was usually 1-2x/week and applicable to most sports. Focus was more on first-step speed, running form, and full body movements (jumps, "suicide" sprints/shuttle runs, etc). I thought it was pretty crazy at the time, still do, but at least it was generic.
These days, neighbors have kids in with pitching coaches at age 8, playing travel league baseball 3+ nights/week, plus 2-3 games/weekend. 9 months of the year. That is REALLY crazy to me. How about some swim team, or soccer, or anything else?
Back when I was in high school (mid-90s), the best athletes were all multi-sport athletes. Football/basketball/track-field was a common combo. No idea if that's true today, but I suspect there's a lot more single-sport athletes at the prep level, which is kinda sad.
This to me is the real problem with over specialization. You have less than gifted parents living out their athletic dreams through their kids but if the parent wasn't that good at the sport there will be correlation with the kid not being that good.
Kids shouldn't specialize in sports too young because there is such a great chance they are not in the right sport at 11.
But to play HS sports for example at most schools, either will do the job. And for most people playing the sport they love in high school will be worth it. Even among pro players, their HS experience is often their best experience.
Better to know the math and not use it for your particular role, than be unqualified because you can’t
A broad base of at least some of stats, linear algebra, and others seems like it would be more useful to most even in STEM to me. I've met folks who were great at those who struggled with geometry and/or calculus, and vice versa. Yet plenty of need for people good at all sorts of those.
Agreed that “memorize rules” is not a great learning experience for students… but it is easy to present.
Point1. Education system which would strive for bear minimum outcome would be something like “not everybody needs addition, cashiers can take care of that”. (You will cut calculus, somebody else will cut multiplication table)
Point2. which iq range education system is optimised for? (Fair discission; though hard politically). Gifted people are random in population. If you teach and test only the bare minimum content, which 99.9% of population passes with flying colors, then gifted are undertrained and also harder to spot, thus under-utilised.
Point3. Memorization is useful. Even if one understands the concept deeply, the “cheat sheet” inside the head (i.e. memorised info) makes things much much faster. Memorization is abit like muscle-memory and or caching.
The problems with rote learning are obvious, and I don’t mean to dismiss them. But I do think the pendulum has swung so far that today, there is excessive and harmful stigma attached to it. Rote learning is a bit of a leap of faith, and it requires trusting that it will be useful in the end (in other words it comes with an upfront cost)… and this is incredibly powerful for taking some academic shortcuts in a limited amount of time.
Now, as an adult, I've noticed how much faster I can get past a lot of tricky issues in my work because the patterns I see a lot are familiar and I can glance over them because I know them by heart and don't get stuck on every little detail.
Stats is a non-starter without calculus. You'd just be doing the "ram a bunch of rules for derivation and integration into your head." method since first principles require calculus. Linear algebra is less reliant on calculus, but the fun stuff still requires calculus. I learned basic linear algebra in high school for what it's worth, just enough to solve systems of linear equations.
Teach it the way you suggest and it would just be another useless math class that people would tune out of instead of grounding it in their everyday life experiences.
("Deriving from first principles" isn't the interesting stuff about calculus, either. And we're not requiring quantum physics to derive gravity from first principles in high school...)
I use linear algebra every day in my work as a graphics programmer. A huge portion of what I do depends on it. While I frequently need calculus it's still a minority of the time.
Derivates obviously show up everywhere: velocity, acceleration, gradients, ...
But it seems to me only the narrow portion relating to physically-based rendering or simulation requires an understanding of integrals. At the moment I can mainly think of how it's used to define continuous functions before introducing the discretization used to actually compute them.
I needed to buy a bunch of precisely cut curved wood for a 2nd level (laser cut wood! So Awesome!), and had to actually re-learn the basics of trig in order to provide the details to have everything made.
In addition, that section of math was the most beautiful thing I've learned in my life.
But I guess his point is rather that most graduates of engineering schools don't even work as engineers at all, which might be true, I don't know (I even have difficulty believing that given the number of people I know who weren't even trained as engineers and are now working as software engineers).
Similarly, is a quarterback running ballistics equations to determine if they can complete a pass? All while running an optimization on the timing of various other players on the field to know if they will have time to make the pass?
Programming is a more interesting issue. If this blog had argued that computer science should be marginalized within software engineering then he might have found something to say (but which has already been said many times by others).
It's not that those jobs are without challenges but they are in other area's, such as requirements gathering and project management.
Ironic that a hypothesis that most people won't need mathematics should require solid statistics to resolve.
Hate to say it, but he looks like a Malcolm Gladwell wannabe with some attention-grabbing book titles. Perfect for a TED talk.
https://www.econlib.org/my-complete-bet-wiki/
EDIT: I would add that Caplan is a working researcher who has made significant and original contributions to economics, where Gladwell is a journalist who interprets the work of scholars like Caplan. If anything, Gladwell is a Caplan wannabe (although I wouldn't characterize either of them in that fashion, I think they have different goals and methods)
The problem I have is that he is looking at averages. In my experience, there are five tiers: counting, arithmetic, algebra, calculus, and higher math. The USA is simultaneously sending more people to the top two and bottom two, with the bottom two going from a small few to a real population segment. I don't even know if the worst public schools teach anything anymore or if they are just pseudo-prisons to capture taxes.
I won't even go into the woeful state of logical and statistical knowledge. I don't know why those aren't core requirements for graduating high school.
And oh, the irony of dismissing the case for mathematics as a thinking aid as “self-serving nonsense” for which the author has “a very low opinion”!
Perhaps if the author had a better attitude towards education and was capable of appreciating the value of mathematics (or more generally, science) in everyday life (to say nothing about the importance of this in democracies), then he would not hold this infantile opinion.
Sorry, not sorry.
Edit: the point is not whether you “use it” when you’re out and about doing your groceries. If you’re judging the value of maths and science by this measure, you’re missing the point, and you’re at such a low level of insight that your perspective is useful to nobody. Rule of thumb: don’t listen to anti-intellectualists.
Caplan has a doctorate, teaches at a university, has written several books and home educated two of his sons to a level where they published academically before they entered university. He’s also more well read in philosophy and literature than average though that is a low bar to clear.
If you don’t think a professor of Mathematics, David Edwards, understands the value of Mathematics why do you think you do?
The author, as decorated as he may be, seemingly fails to appreciate the latter: in his own words, he has a “very low opinion of this self-serving nonsense”. The exercise of fleshing out the argument against the higher-order value of mathematics, you will note, is left as an exercise to the reader.
This is by definition an anti-intellectualist position: defending such a position, which goes contrary to what should be the default position in general, should require a very high burden of proof, not an offhand dismissal.
Directly from the article.
- He doesn't have a high opinion of it
- He makes a vague allusion to some unspecified conflict of interest (?!)
- He argues that first-order skill development is most effective (which may be true, but misses the point).
- He says that higher order value of mathematics is "unsettled".
As he is defending a fundamentally anti-intellectual position, I'm going to need a little more than that.
I’m sure the author has a high opinion of mathematics as an intellectual pursuit. He is a Math professor. That’s separate from his argument, that it has very limited practical use, even to most engineers and others you would assume would be highly selected for finding it useful.
If Mathematics was enormously useful for teaching argument and precise thought in a reliable way Economics would have eaten all the other social sciences already. Economists know far more Math than the others. Math is uncommonly useful but if it was that good at teaching people how to think, if the transfer of learning argument was true, it would not need to be argued. It b would be bloody obvious.
There's this thing called the burden of proof, in philosophy. When you take a difficult position, you must displace this burden. The author has not done so, and it is not my responsibility to show how he is wrong: he has not shown how he is correct.
Anti-intellectual. You keep on saying that the author is anti-intellectual for denigrating the practical value of Math.
What I am saying (or at least what I intend to say) is that he is defending an anti-intellectualist idea/position, which is just a fact by definition.
Saying something is of limited practical value is not an anti-intellectual position. I do not think most people get any practical value out of the chemistry or physics they learn in school.
This is a good example of an anti-intellectual comment or attitude.
Very eloquent phrase, I’m going to steal this!
Lovers of physical exercise, the production of art or combat can sing the praises of their joys and the benefits according thereto. We don’t rearrange the world to force their interests on the world. If we’re going to do it for our interests we should have actual reasoning behind it, not hand waving.
This is not compatible with an inquisitive, academic or intellectual mindset, no matter that he is a professor or that he holds a PhD. He's free to advance arguments in support of this position, but he should be met with extreme skepticism, as he is defending a difficult position.
The world has millions of things to learn about. Why is Math specially subsidized? That’s good argument. That’s not anti-intellectual.
> He's free to advance arguments in support of this position, but he should be met with extreme skepticism, as he is defending a difficult position.
Again, you’re just assuming[edited] your conclusion. You and I and he are people who like ideas and arguing over ideas. He’s arguing that what he teaches isn’t that useful and asking why it gets so many resources. There’s nothing there about the beauty or joy of Math, which I presume he could wax rhapsodic about as a professor. You’re attacking a position that doesn’t appear in article. It’s about the practical value of advanced Math.
First, that's a question, not an argument.
Second, math is special because it's an exact science, as opposed to a natural science. Encompassing the study of "logic", "information", "things that change", "symmetry", "equality" or equivalence... it applies to pretty much everything. In other words it's one of the skills that has the most potential to be widely applicable.
> Again, you’re just aiming your conclusion.
I'm not sure what you mean by "aiming", but I am referring to the burden of proof [1], which the author has not shifted one iota.
[1] https://en.wikipedia.org/wiki/Burden_of_proof_(philosophy)
And we do use that in everyday life. Learning higher maths doesn't help us solve problems we encounter all the time or everyday, at least not directly. But it helps us frame them and our thinking conceptually in a quantitative way, and maybe even sometimes in a qualitative one, which is incredibly helpful generally.
I'm not sure it is a good advice in general. All the quack "doctors" that go against the mainstream medicine are actually degree-earnt doctors, yet are just scams.
GP did not say "against their profession's interests", just their own.
If their kids were kept in normal schools with other kids, they may have published academically a few years later, but a lot of their peers would have had the privilege of studying with them and improved their academic lives because of that.
If you need other students to successfully study, that's either yours or the educational system's failure; but other children shouldn't be dragged down for it.
There is a large number of studies on the positive effects of integration in education, from more extreme cases such as not having special schools for disabled children to the more mundane socioeconomic mixing. The consensus seems to be that the positive academic effect a smart kid gets from being put in some advanced school is trumped by the negative effect on the average kids. European countries with a strong egalitarian bend in their schooling system seem to do much better than the American ones and not just on average.
Europe is not some fantasy-land where all the dreams of the American left come true. Most European countries track students into separate secondary schools based on their demonstrated academic ability.
https://en.wikipedia.org/wiki/List_of_secondary_education_sy...
When I said "European countries with a strong egalitarian bend in their schooling system" I was thinking more about countries like Finland and Scandinavia rather than implying that all European countries are egalitarian.
From what I gather, Finland and Sweden both track students into separate "academic" and "vocational" secondary schools.
I'm not sure; France and Russia and their selective systems, respectively classes préparatoires and elite schools, do not seem to do worse than other European countries.
You're barking up the wrong tree. At worst it's elitist, not anti-intelectual:
"The math myth is the myth that the future of the American economy is dependent upon the masses having higher mathematics skills."
Just doubting the need for higher math skills to be massified, not against their being necessary, perhaps to an elite.
Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.
Also the author is a mathematics professor. He isn’t anti-intellectual.
EDIT Wow. You posted ten angry comments in about 40 minutes. This seems to have hit a nerve.
It doesn't matter who he is: the idea he's advancing is fundamentally an anti-intellectual one. It would be an anti-intellectual idea even if it came out of Einstein's mouth.
Edit in response to edit: sorry if my comments give the appearance of anger, that is sincerely not my intended tone. I am admittedly very passionate about the issue of anti-intellectualism in society.
> Anti-intellectualism is hostility to and mistrust of intellect, intellectuals, and intellectualism, commonly expressed as deprecation of education and philosophy and the dismissal of art, literature, and science as impractical, politically motivated, and even contemptible human pursuits.
https://en.wikipedia.org/wiki/Anti-intellectualism#Education...
For example, let's say there is a group that believes that only those that study rock music are truly intellectual, and holds them in high esteem, while thinking that all others (including physicists, great writers, prophets etc) are lesser minds and not worth paying attention to. This group is not anti-intellectual, as they hold intellectuals in high regard, though having a wildly off-kilter definition of one.
In contrast, many groups view study itself as a bad activity, and those that practice it as haughty, distant, idealistic, and generally not worth listening to - these groups are what it means to be anti-intellectual; the kind that praise "street smarts" far above "book learning". People advocating for math education as the most important form of human thought are nothing alike (though I do agree they are also wrong, in different ways).
I think it's possible for someone to have an anti-intellectual attitude with respect to a subset of domains, however. For example, if I believe there's no real purpose to learning more mathematics, then that's an anti-intellectual attitude towards maths. If I believe there's no real purpose to learning more about music, that's also an anti-intellectual attitude towards the study of music.
(Side note: if I believe there's no use studying homeopathy, well... that's not anti-intellectual, that's just correctly recognizing that homeopathy is not an epistemic domain, ie.: that it is a pseudoscience.)
We (North America) filter children early in the education system on the basis of their ability to do accurate arithmetic. Most jurisdictions don't even really teach math proper until high school. By the time kids get there, they've lost their self-confidence because the sheer tedium and boredom of the curriculum in the early grades.
Personally: Because of my primary and high school math grades I was barred from entering a computer science degree. But I have a 20+ year software engineering career only because I just really loved computers and pushed my way into the industry, and ended up at Google among other places, despite the lack of degree. But it was hard, really hard, to get into these jobs. And there's always been people in those jobs who gave me the sense they didn't think I belonged there with them.
The way we're teaching math to kids is on the whole really awful. I've seen both my kids struggle with it, despite being gifted intellectually and scoring incredibly high on the WISC-V.
The whole process seems geared towards grinding kids down in repetition and struggle.
Meanwhile we have machines to automate arithmetic and most math generally. In general, my sense of the way math is taught these days: it's an insult to the human spirit.
And people who excel in it and struggled through it and "won" have an intrinsic bias in seeing it as the "right" filter and "right" gating for what they're doing today, even when it clearly is not always directly related.
You must not know many people. The problem with trying to categorize K-12 mathematics (or other common, but non-uniform experiences) as good or bad or somewhere in between is precisely that the experience is non-uniform. Even being in the same classes as my twin sister (though sometimes not), we formed some wildly different opinions on different teachers and classes and even classmates. The experience was too different even sitting 5' from each other, let alone when speaking about this with people who attended schools in different parts of the country, or even different parts of the same state or city.
Reading? Now I have access to all these stories! And it's plainly useful in every single other class, pretty much constantly (even math class—go figure, this math book has more words than numbers in it!). I can read subtitles (on anime, say—it's gone pretty mainstream now, for kids). I can read video games without voice acting, and follow them (Breath of the Wild requires lots of reading, for instance). It is constantly, obviously useful, and enables access to lots of things that most kids want to do.
History? Social studies? Largely an application of reading—and hey, look, more stories! Great! Oh and now I understand that movie I watched better, or that game I'm playing.
Science? Well look at that, a ton more reading, and when it's taught halfway right it's like working through a mystery story, even. Fun!
Math? It really seems like you're just doing it for its own sake, and... that's it. The actual facts or what's intended don't matter—that is what it feels like. It's occasionally useful in other classes, but not constantly like reading is, and often that math is pretty simple compared to the stuff you're learning in math class. Kids aren't dumb and recognize quickly that word problems are contrived, pandering, and ridiculous, and meanwhile have a hard time finding applications for any of this even if they go looking for it. Their parents? They probably see them rely on reading (even just signs or directions) 1000x more often than they see them apply math beyond what's taught by 6th grade. Reading? Useful daily. Math? Only occasionally, and it's mostly the simple stuff.
To a student, it feels like math is about as useful as learning some constructed language that everyone's telling them is super important but in their actual experience sure doesn't seem to be—I mean, no adults they know even read, write, or speak it, and they seem to be getting by just fine. It feels like you're being gaslit or having some kind of practical joke played on you. Of course it's hard to care about it, unless you're the sort who just loves mathematical puzzles, like the way some people are really into crossword puzzles, but most people aren't.
I don't have any kind of solution to this, but IMO it's the fundamental problem with math education.
Reading? Yes, the intricacies of clause structure? No, no one uses that besides a few people.
Science? Almost no one I know uses chemistry or biology. I don’t even remember how to balance a chemical equation.
The same arguments against mathematics education can be used for almost anything.
If “Reading” is the example for English education, then “addition, subtraction, multiplication, and division” are the analogs in math, and they’re used by pretty much everyone in day to day life and in other classes. Hell, they’re even used when reading.
The next 6 years? Not so much. Meanwhile, at least the english classes have stories in them, and the kids might not hate all of them.
> Reading? Yes, the intricacies of clause structure? No, no one uses that besides a few people.
Formal grammar is barely taught at all, now. For better or worse. Drilling spelling words has also fallen out of favor.
> The same arguments against mathematics education can be used for almost anything.
They really can't, nowhere near the same degree. If we ignore that and assume more than a tiny minority of kids will give a shit about math if we just present it to them harder or more mathily, then it's not gonna get better.
Do you ever play board games and try to come up with the best strategy? That's math.
And that's not even getting into things that involve numbers…
> Do you ever play board games and try to come up with the best strategy? That's math.
If zero people who aren't math majors think that's true—which is the case—then our 13 years of compulsory math education is to blame for that.
If people manage to do both well enough to get by just fine in life, without any formal math education on it, then one wonders how necessary math education is to effectively do the math that people actually do.
(personally, I'd look in the philosophy section for the former, at least, seeing as all my logic books, except the one specifically about mathematical proofs, are philosophy textbooks, including the ones full of symbols)
I certainly have NEVER used biology at work, and I didn’t do well in it in high school or college.
Math has a much stronger claim to be generically useful. Related fields, such as physics may in fact be seen something worth studing more because it teaches you how to apply math, than for the likelihood that you may end up in a job that requires the physics itself.
My own experience is that math/stats is by far the most useful (non language) topic I studied in school/university, even more so that Computer Science (and I work in IT).
[1]: In my not-professional experience I see that people get lost in math down the road and will struggle in later classes because they have elementary troubles understanding even the basics of what an equation is. For example, I was baffled when one of my classmates argued that the x and y in an equation can't denote the same value.. But I also experienced the same thing when I tried to teach/help math to my sister. Math is simply not a subject like every other where not remembering the previous year's material is not a problem -- it necessarily builds on itself.
I mean I personally think school is overrated, especially with high quality YouTube videos available for everything. But I don't think that is a common notion?
Edit: OK, you know what, why not - yes, abolish schools! They are just prisons for kids.
Seems pretty clear?
Maybe a more useful discussion would be how to measure success and how to determine what is necessary? This "8th grade maths is enough" is a bit too handwavy.
I can actually imagine that there would not have to be as much focus on doing calculations on paper or in one's head, as everybody has a calculator ready at all times these days. As one example.
To start, I think everybody should be able to do financial calculations (like compounding interest), and be able to decipher the charts they are being bombarded with every day in the media.
Other than that I guess I agree with some other posters that it is the people who figure thinks out who are often also good in maths. I am not even sure how to teach that and if traditional maths at school helps.
I don't think that means everything is fine as it is, but also that simply getting rid of subjects because people don't enjoy them may be too much.
Not discussed is that statistics and logic are a much better use of instructional time than calculus for most students.
https://news.ycombinator.com/newsguidelines.html
But I don't believe there was any name-calling, as I attempted to clarify in another comment: the "anti-intellectualist" label was intended to refer to the idea expressed in the article, not to the person who wrote it. (And even in the case where the label were attached to a person, it's not really an insult: I mean, it's definitely not a compliment, but to be clear, it says absolutely nothing of the person's intelligence, and rather speaks to their attitude towards "the pursuit of knowledge" in the abstract.)
I recognize that my intended meaning was not as clear as it should have been in the root comment, but I hope I have clarified this anywhere else. This was careless phrasing, but there was no ill intent.
"When disagreeing, please reply to the argument instead of calling names. 'That is idiotic; 1 + 1 is 2, not 3' can be shortened to '1 + 1 is 2, not 3." - https://news.ycombinator.com/newsguidelines.html
Thanks for the nice reply in any case.
> I must study Politicks and War that my sons may have liberty to study Mathematicks and Philosophy. My sons ought to study Mathematicks and Philosophy, Geography, natural History, Naval Architecture, navigation, Commerce and Agriculture, in order to give their Children a right to study Painting, Poetry, Musick, Architecture, Statuary, Tapestry and Porcelaine.
Somewhere along the line it seems we got stuck at what Adams wanted his sons to study, and never really moved on to what he hoped his grandchildren could study.
A big issue with the motivation to learn math is that it doesn't tell a compelling story for students for many years. But perhaps that could change.
Solving one scary calculus problem thrown at me as an undergrad - an integral - resulted in a solution which broke it into 16 separate integrals. Days of algebra were involved.
* If daily user growth is increasing linearly, total user count is exponential over time.
* If I have a radially symmetric shape it's center of gravity is going to be centered on the axis of symmetry - if the density is uniform. But if the density isn't uniform, where will the center of balance shift?
This is single dimension and multi-dimensional calculus, respectively. I bet most engineers use at least the former, at an intuitive level, on a regular basis.
Exponential growth would be when daily growth is proportional to the current number of users, which is different.
cracking me up here
When I didn't understand calculus I thought that was pretty unessential for being a good programmer. Strange thing though, after learning calculus I find myself using to help understand or model problems a couple of times a week at least.
When I didn't understand linear algebra I didn't see how useful it was for anything outside of the obvious niches like graphics programming. But once I learned it I was able to reduce a remarkable number of complex problems to literally 1-3 lines of code.
Even when I knew statistics, I didn't see much use in all that fancy advanced theoretical stuff. Now I find one of the most useful, practical tools when modeling problems is the Laplace approximation, and knowing the Cramer-Rao bound let's me realize the limits of the results I get from this.
Perhaps most famously, the great mathematician G.H. Hardy, in 1940, discussed how permanently impractical and "useless" number theory was (to be clear, he cited this as a virtue of number theory) in his Mathematicians Apology. Just a few years later Alan Turing (and several others) realized that number theory was the key to securing (and undermining) cryptographic systems.
The lesson for me has been that programmers (and I'm sure many other professions) have a habit of finding good uses for the math they've learned.
The welfare cliff and similar issues wouldn't happen if people could communicate using continuous functions.
That seems incredible to me. I guess that means I live in a bubble.
Surely they would have learned concepts of compound interest by then?
Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.
I studied economics and computer science at a top-20 university. I did a grand total of one course in the math department (1st semester calculus). Everything else was specific to my areas of study. I don’t recall anybody being forced to take high level mathematics outside of engineering programs. Even within the hard sciences, the requirement was usually something like “take 2 semesters of math, plus statistics.” Or something like that.
Would the author also argue that since I don’t use history at my job I should never have taken government, architectural history, or a survey of western religions? Those all contribute to my understanding of the world around me. As does math.
If his point is we need more generalists with a broad understanding of the world, including social sciences and history, that could be a valid point, but it’s not the one he made.
This is very much in line with Caplan's prior views. It holds true to a degree, but probably isn't universal.
Sure -- most engineers shouldn't use advanced math, because that's easy to screw up. But either they are aware that their linear models came from some more complex equations that have been simplified, or they are living in a magic demon-haunted world where equations are just runes handed down by mystic sages. Hopefully it is the former!
Teaching engineers the hard math serves dual purposes -- it helps them understand where their models come from, and somebody is going to have to invent the models. They are one population that has already shown at least some mathematical aptitude.